PhyslibAlpha.CondensedMatter.IsingModel.Basic

The function maps a boolean value $b$ to a real number $s \in \{1, -1\}$, where $s = 1$ if $b$ is true and $s = -1$ if $b$ is false, representing the numerical value of a spin in the Ising model.

Given a finite set of sites $X$ in an Ising model, the function returns the set $X \cup \bigcup_{i \in X} \text{nbrs}(i)$, where $\text{nbrs}(i)$ is the set of sites that interact with site $i$. This set represents the region $X$ together with all sites that have an interaction term with any site within $X$.

Let $X$ be a finite set of sites in an Ising model. Let $X^+$ denote the neighborhood of $X$, which is the set consisting of $X$ and all sites that interact with at least one site in $X$. Then $X$ is a subset of its neighborhood, written as $X \subseteq X^+$.

For any finite sets of sites $X$ and $Y$ in an Ising model, if $X \subseteq Y$, then $X^+ \subseteq Y^+$, where $X^+$ denotes the neighborhood of $X$ defined as $X \cup \bigcup_{i \in X} \text{nbrs}(i)$.

For a finite set of sites $X$ in an Ising model and any site $i \in X$, the set of neighbors of $i$, denoted by $\text{nbrs}(i)$, is a subset of the neighborhood $X^+$, where $X^+$ is defined as $X \cup \bigcup_{j \in X} \text{nbrs}(j)$.

For an Ising model with exchange couplings $J_{ij}$ and longitudinal fields $h_i$, given a finite region of sites $Y$ and a configuration $\sigma: S \to \{\text{true}, \text{false}\}$ assigned to all sites in the lattice $S$, the energy $E_Y(\sigma)$ is defined as: $$E_Y(\sigma) = -\frac{1}{2} \sum_{i \in Y} \sum_{j \in Y} J_{ij} s_i s_j - \sum_{i \in Y} h_i s_i$$ where $s_i = 1$ if $\sigma(i)$ is true and $s_i = -1$ if $\sigma(i)$ is false. This calculation uses free boundary conditions, meaning only the spins and interactions localized within the region $Y$ are included.

Given a set of sites $S$ and a finite subset $Y \subseteq S$, let $c: Y \to \{\text{true}, \text{false}\}$ be a spin configuration restricted to $Y$. This function extends $c$ to a global configuration $\sigma: S \to \{\text{true}, \text{false}\}$ on all sites by setting $$\sigma(s) = \begin{cases} c(s) & \text{if } s \in Y \\ \text{true} & \text{if } s \notin Y \end{cases}$$

For an Ising model on a lattice $S$ with exchange couplings $J_{ij}$ and longitudinal fields $h_i$, given a finite region $Y \subseteq S$ and a local configuration $c: Y \to \{\text{true}, \text{false}\}$ assigned to the sites in $Y$, the energy $E_Y(c)$ is defined as: $$E_Y(c) = -\frac{1}{2} \sum_{i \in Y} \sum_{j \in Y} J_{ij} s_i s_j - \sum_{i \in Y} h_i s_i$$ where $s_k = 1$ if $c(k)$ is true and $s_k = -1$ if $c(k)$ is false. This definition corresponds to free boundary conditions, as the summation only considers interactions between sites both of which are contained within the region $Y$.

Let $Y$ be a finite set of sites in a lattice $S$, and let $\sigma, \sigma': S \to \{\text{true}, \text{false}\}$ be two spin configurations. If the configurations agree on all sites within $Y$ (i.e., $\sigma(s) = \sigma'(s)$ for all $s \in Y$), then the energy $E_Y$ calculated for the region $Y$ is the same for both configurations: $$E_Y(\sigma) = E_Y(\sigma')$$ This reflects that the energy of a region $Y$ under free boundary conditions depends only on the spins located within $Y$.

Given two spin configurations $\sigma, \sigma'$ and a pair of lattice sites $i, j$, this function calculates the contribution of the interaction between $i$ and $j$ to the total energy difference $E(\sigma) - E(\sigma')$. The value is defined as: $-\frac{1}{2} J_{ij} (s_i s_j - s'_i s'_j)$ where $J_{ij}$ is the exchange coupling between sites $i$ and $j$, and $s_k, s'_k \in \{1, -1\}$ are the numerical values of the spins at site $k$ for configurations $\sigma$ and $\sigma'$ respectively.

Given an Ising model with longitudinal fields $h_i$ at each site $i \in S$, let $\sigma$ and $\sigma'$ be two spin configurations mapping sites to boolean values. Let $s_i$ and $s'_i$ denote the numerical spin values ($\pm 1$) corresponding to $\sigma(i)$ and $\sigma'(i)$ respectively. The function calculates the contribution of the field at site $i$ to the energy difference between the two configurations, defined as $h_i (s_i - s'_i)$.

For any finite region of sites $Y$ and any two configurations $\sigma, \sigma' : S \to \{\text{true}, \text{false}\}$, the difference between the energies $E_Y(\sigma)$ and $E_Y(\sigma')$ is equal to the sum over all pairs $i, j \in Y$ of the pair interaction differences minus the sum over all sites $i \in Y$ of the longitudinal field differences: $$E_Y(\sigma) - E_Y(\sigma') = \sum_{i \in Y} \sum_{j \in Y} \text{pairDiff}(\sigma, \sigma', i, j) - \sum_{i \in Y} \text{fieldDiff}(\sigma, \sigma', i)$$ where $\text{pairDiff}(\sigma, \sigma', i, j) = -\frac{1}{2} J_{ij} (s_i s_j - s'_i s'_j)$ and $\text{fieldDiff}(\sigma, \sigma', i) = h_i (s_i - s'_i)$, with $s_k, s'_k \in \{1, -1\}$ being the spin values associated with $\sigma$ and $\sigma'$ respectively.

Let $X$ be a finite set of sites in an Ising model $I$ on a lattice $S$. Let $\sigma$ and $\sigma'$ be two spin configurations such that $\sigma_s = \sigma'_s$ for all sites $s \notin X$. Let $X^+ = X \cup \bigcup_{k \in X} \text{nbrs}(k)$ denote the neighborhood of $X$, where $\text{nbrs}(k)$ is the set of sites interacting with $k$. For any two sites $i, j \in S$, if at least one of them is not in $X^+$ (i.e., $i \notin X^+$ or $j \notin X^+$), then the pair contribution to the energy difference between $\sigma$ and $\sigma'$ is zero: $-\frac{1}{2} J_{ij} (s_i s_j - s'_i s'_j) = 0$ where $J_{ij}$ is the exchange coupling and $s_k, s'_k \in \{1, -1\}$ are the numerical spin values.

In an Ising model $I$ on a set of sites $S$, let $X \subseteq S$ be a finite subset of sites. Suppose $\sigma$ and $\sigma'$ are two spin configurations that agree on all sites outside of $X$ (i.e., $\sigma(s) = \sigma'(s)$ for all $s \notin X$). Then, for any site $i \notin X$, the contribution of the longitudinal field $h_i$ to the energy difference between the two configurations at that site is zero: $h_i(s_i - s'_i) = 0$, where $s_i$ and $s'_i$ are the numerical spin values ($\pm 1$) corresponding to $\sigma(i)$ and $\sigma'(i)$.

Consider an Ising model on a lattice $S$. Let $X$ and $Y$ be finite subsets of sites such that the neighborhood of $X$, denoted by $X^+$, is contained in $Y$ (where $X^+ = X \cup \bigcup_{i \in X} \text{nbrs}(i)$). Suppose $\sigma$ and $\sigma'$ are two spin configurations on $S$ that are identical outside of $X$, i.e., $\sigma(s) = \sigma'(s)$ for all $s \notin X$. Then the difference between the energies of $\sigma$ and $\sigma'$ in the region $Y$ is equal to the difference of their energies calculated in the neighborhood $X^+$: $$E_Y(\sigma) - E_Y(\sigma') = E_{X^+}(\sigma) - E_{X^+}(\sigma')$$ where $E_R(\sigma)$ denotes the energy of configuration $\sigma$ restricted to the region $R$.

Consider an Ising model $I$ on a lattice $S$. Let $X$ and $Y$ be finite subsets of sites such that $X \subseteq Y$ and the neighborhood of $X$ (the set of sites $X$ along with their interacting neighbors), denoted by $X^+$, is also contained in $Y$ ($X^+ \subseteq Y$). Let $c$ and $c'$ be two spin configurations on the region $Y$ (functions from $Y$ to $\{\text{true}, \text{false}\}$) that agree on all sites in $Y \setminus X$ (i.e., $c(y) = c'(y)$ for all $y \in Y$ such that $y \notin X$). Then the difference in the energies of these configurations in region $Y$ is equal to the difference of their energies calculated in the neighborhood $X^+$: $$E_Y(c) - E_Y(c') = E_{X^+}(c|_{X^+}) - E_{X^+}(c'|_{X^+})$$ where $c|_{X^+}$ denotes the restriction of the configuration $c$ to the subset $X^+$.

For an Ising model with inverse temperature $\beta$, the Gibbs weight of a spin configuration $c$ in a finite region $Y$ is defined as $e^{-\beta E_Y(c)}$, where $E_Y(c)$ denotes the energy of the configuration in region $Y$.

Let $I$ be an Ising model on a set of sites $S$. For any inverse temperature $\beta \in \mathbb{R}$, any finite subset of sites $Y \subseteq S$, and any spin configuration $c: Y \to \{\text{true, false}\}$, the Gibbs weight $w_\beta(Y, c) = e^{-\beta E_Y(c)}$ is strictly positive: $$w_\beta(Y, c) > 0$$ where $E_Y(c)$ is the energy of the configuration $c$ in the region $Y$.

For an Ising model with inverse temperature $\beta$, the partition function $Z_Y(\beta)$ for a finite region $Y$ is defined as the sum of the Gibbs weights $e^{-\beta E_Y(c)}$ over all possible spin configurations $c: Y \to \{\text{false, true}\}$.

For an Ising model $I$ on a set of sites $S$, let $Y \subseteq S$ be a finite subset of sites and $\beta \in \mathbb{R}$ be the inverse temperature. The partition function $Z_Y(\beta)$ is strictly positive: $$Z_Y(\beta) > 0$$ where $Z_Y(\beta) = \sum_{c} e^{-\beta E_Y(c)}$ is the sum of the Gibbs weights over all possible spin configurations $c: Y \to \{\text{false, true}\}$ in the region $Y$.